Answer
$\iint_S curl F \cdot dS=0$
Work Step by Step
Divergence Theorem: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, $S$ shows a closed surface. The region $E$ is inside that surface.
We have $div F=\dfrac{\partial P}{\partial x}+\dfrac{\partial q}{\partial y}+\dfrac{\partial r}{\partial z}$
Here, we have $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV $
or, $\iint_S curl[\overrightarrow{F}\cdot d\overrightarrow{S}]=\iiint_E div (curl F)dV $
This implies that $\iint_S curl\overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_E div (curl F)dV=\iiint_E (0) dV $
Thus, we have $div (curl F)=0$
and $\iint_S curl F \cdot dS=0$ (Verified)
